Boolean algebras with no rigid or homogeneous factors

Author:
Petr Štěpánek

Journal:
Trans. Amer. Math. Soc. **270** (1982), 131-147

MSC:
Primary 06E05; Secondary 03E40

DOI:
https://doi.org/10.1090/S0002-9947-1982-0642333-3

MathSciNet review:
642333

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Abstract: A simple construction of Boolean algebras with no rigid or homogeneous factors is described. It is shown that for every uncountable cardinal there are isomorphism types of Boolean algebras of power with no rigid or homogeneous factors. A similar result is obtained for complete Boolean algebras for certain regular cardinals. It is shown that every Boolean algebra can be completely embedded in a complete Boolean algebra with no rigid or homogeneous factors in such a way that the automorphism group of the smaller algebra is a subgroup of the automorphism group of the larger algebra. It turns out that the cardinalities of antichains in both algebras are the same. It is also shown that every -distributive complete Boolean algebra can be completely embedded in a -distributive complete Boolean algebra with no rigid or homogeneous factors.

**[1]**William B. Easton,*Powers of regular cardinals*, Ann. Math. Logic**1**(1970), 139–178. MR**0269497**, https://doi.org/10.1016/0003-4843(70)90012-4**[2]**R. Engelking and M. Karłowicz,*Some theorems of set theory and their topological consequences*, Fund. Math.**57**(1965), 275–285. MR**0196693****[3]**P. Erdős and R. Rado,*Intersection theorems for systems of sets*, J. London Math. Soc.**35**(1960), 85–90. MR**0111692**, https://doi.org/10.1112/jlms/s1-35.1.85**[4]**Ronald B. Jensen and Håvard Johnsbråten,*A new construction of a non-constructible Δ₃¹ subset of 𝜔*, Fund. Math.**81**(1974), 279–290. Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, IV. MR**0419229****[5]**S. Koppelberg,*A lattice structure on the isomorphism types of complete Boolean algebras*, Freie Universität Berlin, 1979, 41 pp.,**preprint**.**[6]**Saul A. Kripke,*An extension of a theorem of Gaifman-Hales-Solovay*, Fund. Math.**61**(1967), 29–32. MR**0220641****[7]**Kenneth McAloon,*Consistency results about ordinal definability*, Ann. Math. Logic**2**(1970/71), no. 4, 449–467. MR**0292670**, https://doi.org/10.1016/0003-4843(71)90005-2**[8]**R. McKenzie and J. D. Monk,*On automorphism groups of Boolean algebras*, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, North-Holland, Amsterdam, 1975, pp. 951–988. Colloq. Math. Soc. János Bolyai, Vol. 10. MR**0376476****[9]**J. D. Monk and W. Rassbach,*The number of rigid Boolean algebras*, Algebra Universalis**9**(1979), no. 2, 207–210. MR**523934**, https://doi.org/10.1007/BF02488031**[10]**S. Roguski,*Extensions of models for ZFC to models for 𝑍𝐹+𝑉=𝐻𝑂𝐷 with applications*, Set theory and hierarchy theory (Proc. Second Conf., Bierutowice, 1975), Springer, Berlin, 1976, pp. 241–247. Lecture Notes in Math., Vol. 537. MR**0429554****[11]**Matatyahu Rubin,*On the reconstruction of Boolean algebras from their automorphism groups*, Handbook of Boolean algebras, Vol. 2, North-Holland, Amsterdam, 1989, pp. 547–606. MR**991603****[12]**Saharon Shelah,*Why there are many nonisomorphic models for unsuperstable theories*, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 259–263. MR**0422015****[13]**Petr Štěpánek,*Cardinal collapsing and ordinal definability*, J. Symbolic Logic**43**(1978), no. 4, 635–642. MR**518669**, https://doi.org/10.2307/2273502

Petr Štěpánek,*Cardinals in the inner model 𝐻𝑂𝐷*, Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Math., vol. 669, Springer, Berlin, 1978, pp. 437–454. MR**520199****[14]**Petr Štěpánek and Bohuslav Balcar,*Embedding theorems for Boolean algebras and consistency results on ordinal definable sets*, J. Symbolic Logic**42**(1977), no. 1, 64–76. MR**0498122**, https://doi.org/10.2307/2272320

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0642333-3

Keywords:
Boolean algebras with no rigid or homogeneous factors,
automorphisms,
embeddings and isomorphism types of BA's,
chain conditions,
distributivity

Article copyright:
© Copyright 1982
American Mathematical Society